Question: Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}\]and $\mathbf{A}^2 = \mathbf{I}.$  Find the largest possible value of $a + b + c.$
We have that
\begin{align*}
\mathbf{A}^2 &= \frac{1}{25} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix} \\
&= \frac{1}{25} \begin{pmatrix} 9 + ab & -3a + ac \\ -3b + bc & ab + c^2 \end{pmatrix}.
\end{align*}Thus, $9 + ab = ab + c^2 = 25$ and $-3a + ac = -3b + bc = 0.$

From $9 + ab = ab + c^2 = 25,$ $ab = 16$ and $c^2 = 9,$ so $c = \pm 3.$

If $c = -3,$ then $-6a = -6b = 0,$ so $a = b = 0.$  But then $ab = 0,$ contradiction, so $c = 3.$  Thus, any values of $a,$ $b,$ and $c$ such that $ab = 16$ and $c = 3$ work.

We want to maximize $a + b + c = a + \frac{16}{a} + 3.$  Since $a$ is an integer, $a$ must divide 16.  We can then check that $a + \frac{16}{a} + 3$ is maximized when $a = 1$ or $a = 16,$ which gives a maximum value of $\boxed{20}.$